File size: 4,514 Bytes
92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import OAI.Dynamics.StandardMap.FastBridge
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
open MeasureTheory Set Filter Metric
open scoped ENNReal Topology Classical
namespace StandardMapEntropy
lemma strict_markov_real {X : Type*} [MeasurableSpace X] (μ : Measure X)
(H : X → ℝ) (hi : Integrable H μ) (h0 : ∀ x,0≤H x) (h : ℝ) (hh : 0<h) :
μ {x | h<H x} ≤ ENNReal.ofReal ((∫ x,H x ∂μ)/h) := by
have hb := meas_ge_le_lintegral_div hi.aestronglyMeasurable.aemeasurable.ennreal_ofReal
(ENNReal.ofReal_ne_zero_iff.mpr hh) ENNReal.ofReal_ne_top
rw [←ofReal_integral_eq_lintegral_ofReal hi (Eventually.of_forall h0)] at hb
have hs : {x | h<H x} ⊆ {x | ENNReal.ofReal h≤ENNReal.ofReal (H x)} := fun x hx => ENNReal.ofReal_le_ofReal hx.le
exact (measure_mono hs).trans (hb.trans_eq (ENNReal.ofReal_div_of_pos hh).symm)
lemma real_integral_bound_of_lintegral {X : Type*} [MeasurableSpace X] (μ : Measure X)
(W : X → ℝ) (hi : Integrable W μ) (h0 : ∀ x,0≤W x) (E : Set X)
(C : ℝ) (hC : 0≤C) (hb : (∫⁻ x in E,ENNReal.ofReal (W x) ∂μ)≤ENNReal.ofReal C) :
(∫ x in E,W x ∂μ) ≤ C := by
rw [←ofReal_integral_eq_lintegral_ofReal hi.restrict (Eventually.of_forall h0)] at hb
exact (ENNReal.ofReal_le_ofReal_iff hC).mp hb
noncomputable def bridgeRealConstant : ℝ := fastBridgeConstant.toReal
lemma bridgeRealConstant_nonneg : 0≤bridgeRealConstant := ENNReal.toReal_nonneg
lemma actual_fast_bridge_real (k : ℝ) (hc : BridgeScalarControl k)
(N r : ℕ) (hN : 100 ≤ N) (hr : (r:ℝ) ≤ (N:ℝ)/1000)
{Rw Rh : ℕ} (aw : Fin Rw → ℤ) (nw : Fin Rw → ℕ) (Fw : (Fin Rw → ℝ) → ℝ)
(ah : Fin Rh → ℤ) (nh : Fin Rh → ℕ) (Fh : (Fin Rh → ℝ) → ℝ)
(hnw : ∀ j, 0 < nw j) (hnh : ∀ j, 0 < nh j) (L : NNReal)
(hFw : LipschitzWith L Fw) (hFh : LipschitzWith L Fh)
(hlenw : ∀ j, nw j ≤ 2*r) (hlenh : ∀ j, nh j ≤ 2*r)
(hwinw : ∀ j l, 1 ≤ l → l ≤ nw j → (aw j+(l:ℤ)-1).natAbs ≤ r)
(hwinh : ∀ j l, 1 ≤ l → l ≤ nh j → (ah j+(l:ℤ)-1-(N:ℤ)).natAbs ≤ r)
(hW0 : ∀ z, 0 ≤ shortfallObservation k aw nw Fw z)
(hH0 : ∀ z, 0 ≤ shortfallObservation k ah nh Fh z)
(hlog : 1 ≤ Real.log (growthBase k)) (hLM : 16*Real.pi*L ≤ growthBase k)
(h : ℝ) (hh : 0<h) (hδ : growthBase k^(-(1/10:ℝ)*(N:ℝ)) < h/2) :
(∫ z in {z | torusFastBridge k z N} ∩ {z | h < shortfallObservation k ah nh Fh z},
shortfallObservation k aw nw Fw z ∂area) ≤
bridgeRealConstant*((∫ z,shortfallObservation k aw nw Fw z ∂area)+
growthBase k^(-(1/10:ℝ)*(N:ℝ)))*((∫ z,shortfallObservation k ah nh Fh z ∂area)/(h/2)) := by
have hW := continuous_shortfall_observation k hc.nonneg aw nw Fw hFw.continuous
have hH := continuous_shortfall_observation k hc.nonneg ah nh Fh hFh.continuous
have hiW := hW.integrable_of_hasCompactSupport (μ := area) (HasCompactSupport.of_compactSpace _)
have hiH := hH.integrable_of_hasCompactSupport (μ := area) (HasCompactSupport.of_compactSpace _)
have hb := actual_fast_bridge_product k hc N r hN hr aw nw Fw ah nh Fh hnw hnh L hFw hFh
hlenw hlenh hwinw hwinh hW0 hlog hLM h hδ
have hd0 : 0≤growthBase k^(-(1/10:ℝ)*(N:ℝ)) := Real.rpow_nonneg (by linarith [growthBase_ge_four k hc.nonneg]) _
have hiW0 := integral_nonneg_of_ae (μ := area) (Eventually.of_forall hW0)
have hiH0 := integral_nonneg_of_ae (μ := area) (Eventually.of_forall hH0)
apply real_integral_bound_of_lintegral area _ hiW hW0 _ _
(mul_nonneg (mul_nonneg bridgeRealConstant_nonneg (add_nonneg hiW0 hd0)) (div_nonneg hiH0 (by linarith)))
have hm := strict_markov_real area _ hiH hH0 (h/2) (by linarith)
apply (hb.trans (mul_le_mul_of_nonneg_left hm bot_le)).trans_eq
change fastBridgeConstant*((∫⁻ z,ENNReal.ofReal (shortfallObservation k aw nw Fw z) ∂area)+ENNReal.ofReal (growthBase k^(-(1/10:ℝ)*(N:ℝ))))*ENNReal.ofReal ((∫ z,shortfallObservation k ah nh Fh z ∂area)/(h/2))=_
rw [←ofReal_integral_eq_lintegral_ofReal (f := shortfallObservation k aw nw Fw) hiW (Eventually.of_forall hW0),
←ENNReal.ofReal_add hiW0 hd0]
conv_lhs => rw [show fastBridgeConstant=ENNReal.ofReal bridgeRealConstant from (ENNReal.ofReal_toReal fastBridgeConstant_ne_top).symm]
rw [
←ENNReal.ofReal_mul bridgeRealConstant_nonneg,←ENNReal.ofReal_mul (mul_nonneg bridgeRealConstant_nonneg (add_nonneg hiW0 hd0))]
end StandardMapEntropy
end OAI
|